EXACT SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH SPHERICALLY SYMMETRIC OCTIC POTENTIAL

2012 ◽  
Vol 27 (20) ◽  
pp. 1250112 ◽  
Author(s):  
DAVIDS AGBOOLA ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Closed Form ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Spherically Symmetric ◽  
Algebraic Equations ◽  
Potential Parameters

We present exact solutions of the Schrödinger equation with spherically symmetric octic potential. We give closed-form expressions for the energies and the wave functions as well as the allowed values of the potential parameters in terms of a set of algebraic equations.

EXACT WAVE FUNCTIONS AND ENERGIES OF A NON-RELATIVISTIC FREE QUANTUM PARTICLE ON THE SURFACE OF A DEGENERATE TORUS

2004 ◽  
Vol 19 (23) ◽  
pp. 1759-1766 ◽  
Author(s):  
AXEL SCHULZE-HALBERG
Keyword(s):  
Schrödinger Equation ◽  
Closed Form ◽  
Schrodinger Equation ◽  
Quantum Particle ◽  
Azimuthal Angle ◽  
Polar Angle ◽  
Wave Functions ◽  
Exactly Solvable

We study the non-relativistic Schrödinger equation for a free quantum particle constrained to the surface of a degenerate torus, parametrized by its polar and azimuthal angle. On restricting to wave functions that depend on the polar angle only, the Schrödinger equation becomes exactly-solvable. We compute its physical solutions (continuous, normalizable and 2π-periodic) and the associated energies in closed form.

scholarly journals SOLVING THE SCHRÖDINGER EQUATION FOR BOUND STATES WITH MATHEMATICA 3.0

1999 ◽  
Vol 10 (04) ◽  
pp. 607-619 ◽  
Author(s):  
WOLFGANG LUCHA ◽  
FRANZ F. SCHÖBERL
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Interaction Potential ◽  
Schrodinger Equation ◽  
Bound States ◽  
Energy Eigenvalue ◽  
Wave Functions ◽  
Spherically Symmetric ◽  
Method Of Solution ◽  
Numerical Integration Procedure

Using Mathematica 3.0, the Schrödinger equation for bound states is solved. The method of solution is based on a numerical integration procedure together with convexity arguments and the nodal theorem for wave functions. The interaction potential has to be spherically symmetric. The solving procedure is simply defined as some Mathematica function. The output is the energy eigenvalue and the reduced wave function, which is provided as an interpolated function (and can thus be used for the calculation of, e.g., moments by using any Mathematica built-in function) as well as plotted automatically. The corresponding program schroedinger.nb can be obtained from [email protected].

scholarly journals A Numerical Approach for the Solution of Schrödinger Equation With Pseudo-Gaussian Potentials

2015 ◽  
Vol 58 (1) ◽  
pp. 1-6
Author(s):  
Theodor-Felix Iacob ◽  
Marina Lute ◽  
Felix Iacob
Keyword(s):  
Numerical Methods ◽  
Power Series ◽  
Schrödinger Equation ◽  
Exact Solutions ◽  
Schrodinger Equation ◽  
Numerical Approach ◽  
Wave Functions ◽  
Gaussian Potential ◽  
Convergence Of Series ◽  
Infinite Power

Abstract The Schrödinger equation with pseudo-Gaussian potential is investigated. The pseudo-Gaussian potential can be written as an infinite power series. Technically, by an ansatz to the wave-functions, exact solutions can be found by analytic approach [12]. However, to calculate the solutions for each state, a condition that will stop the series has to be introduced. In this way the calculated energy values may suffer modifications by imposing the convergence of series. Our presentation, based on numerical methods, is to compare the results with those obtained in the analytic case and to determine if the results are stable under different stopping conditions.

scholarly journals A New Constructive Method for Solving the Schrödinger Equation

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1879
Author(s):  
Kazimierz Rajchel
Keyword(s):  
Energy Spectrum ◽  
Schrödinger Equation ◽  
Exact Solutions ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Parametric Family ◽  
Constructive Method ◽  
One Dimensional ◽  
Parametric Solutions ◽  
Ricatti Equation

In this paper, a new method for the exact solution of the stationary, one-dimensional Schrödinger equation is proposed. Application of the method leads to a three-parametric family of exact solutions, previously known only in the limiting cases. The method is based on solutions of the Ricatti equation in the form of a quadratic function with three parameters. The logarithmic derivative of the wave function transforms the Schrödinger equation to the Ricatti equation with arbitrary potential. The Ricatti equation is solved by exploiting the particular symmetry, where a family of discrete transformations preserves the original form of the equation. The method is applied to a one-dimensional Schrödinger equation with a bound states spectrum. By extending the results of the Ricatti equation to the Schrödinger equation the three-parametric solutions for wave functions and energy spectrum are obtained. This three-parametric family of exact solutions is defined on compact support, as well as on the whole real axis in the limiting case, and corresponds to a uniquely defined form of potential. Celebrated exactly solvable cases of special potentials like harmonic oscillator potential, Coulomb potential, infinite square well potential with corresponding energy spectrum and wave functions follow from the general form by appropriate selection of parameters values. The first two of these potentials with corresponding solutions, which are defined on the whole axis and half axis respectively, are achieved by taking the limit of general three-parametric solutions, where one of the parameters approaches a certain, well-defined value.

Quantum Boxes, Stringed Instruments

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Energy Level ◽  
Schrödinger Equation ◽  
Quantum System ◽  
Schrodinger Equation ◽  
Probability Distributions ◽  
Wave Functions ◽  
Energy Level Diagram ◽  
One Dimensional ◽  
Stringed Instruments ◽  
Symmetry Properties

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

Sign in / Sign up

Export Citation Format

Share Document